Homogenization of the Allen-Cahn equation with periodic mobility

TL;DR

This paper studies the large-scale behavior of interfaces in the Allen-Cahn equation with periodic mobility, showing they follow an effective mean curvature flow despite challenges with ill-posed cell problems.

Contribution

It introduces new methods to handle ill-posed cell problems in the homogenization of anisotropic Allen-Cahn equations with periodic coefficients.

Findings

Interfaces follow an effective mean curvature flow.

New techniques address ill-posed cell problems.

Large-scale interface behavior is characterized by homogenized equations.

Abstract

We analyze the sharp interface limit for the Allen-Cahn equation with an anisotropic, spatially periodic mobility coefficient and prove that the large-scale behavior of interfaces is determined by mean curvature flow with an effective mobility. Formally, the result follows from the asymptotics developed by Barles and Souganidis for bistable reaction-diffusion equations with periodic coefficients. However, we show that the corresponding cell problem is actually ill-posed when the normal direction is rational. To circumvent this issue, a number of new ideas are needed, both in the construction of mesoscopic sub- and supersolutions controlling the large-scale behavior of interfaces and in the proof that the interfaces obtained in the limit are actually described by the effective equation.